Theorem nnssn0s 28327

Description: The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
nnssn0s	⊢ ℕs ⊆ ℕ0s

Proof of Theorem nnssn0s

Step	Hyp	Ref	Expression
1		df-nns 28321	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
2		difss 4077	. 2 ⊢ (ℕ0s ∖ { 0s }) ⊆ ℕ0s
3	1, 2	eqsstri 3969	1 ⊢ ℕs ⊆ ℕ0s

Syntax hints:   ∖ cdif 3887   ⊆ wss 3890  {csn 4568   0_s c0s 27811  ℕ_0scn0s 28318  ℕ_scnns 28319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-dif 3893  df-ss 3907  df-nns 28321

This theorem is referenced by:  nnssno  28328  nnn0s  28333  nnn0sd  28334  nnsgt0  28345